Download Interference Energy Spectrum of the Infinite Square Well

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Interference Energy Spectrum of the Infinite
Square Well
Mordecai Waegell *, Yakir Aharonov and Taylor Lee Patti
Institute for Quantum Studies, Chapman University, Orange, CA 92866, USA; [email protected] (Y.A.);
[email protected] (T.L.P.)
* Correspondence: [email protected]
Academic Editors: Gregg Jaeger and Andrei Khrennikov
Received: 26 February 2016; Accepted: 13 April 2016; Published: 19 April 2016
Abstract: Certain superposition states of the 1-D infinite square well have transient zeros at
locations other than the nodes of the eigenstates that comprise them. It is shown that if an
infinite potential barrier is suddenly raised at some or all of these zeros, the well can be split into
multiple adjacent infinite square wells without affecting the wavefunction. This effects a change
of the energy eigenbasis of the state to a basis that does not commute with the original, and a
subsequent measurement of the energy now reveals a completely different spectrum, which we
call the interference energy spectrum of the state. This name is appropriate because the same
splitting procedure applied at the stationary nodes of any eigenstate does not change the measurable
energy of the state. Of particular interest, this procedure can result in measurable energies that are
greater than the energy of the highest mode in the original superposition, raising questions about
the conservation of energy akin to those that have been raised in the study of superoscillations.
An analytic derivation is given for the interference spectrum of a given wavefunction Ψ(x, t) with
N known zeros located at points si = (xi , ti ). Numerical simulations were used to verify that
a barrier can be rapidly raised at a zero of the wavefunction without significantly affecting it.
The interpretation of this result with respect to the conservation of energy and the energy-time
uncertainty relation is discussed, and the idea of alternate energy eigenbases is fleshed out.
The question of whether or not a preferred discrete energy spectrum is an inherent feature of a
particle’s quantum state is examined.
Keywords: energy spectrum; uncertainty principle; energy conservation; superoscillation;
frequency conversion; infinite square well
1. Introduction
The purpose of this letter is to show that it is possible, in principle, to measure alternate energy
eigenbases of a given superposition state of the infinite square well and that the highest energy
eigenstate in a given superposition may have a different energy in different bases.
The origin of this idea goes back to the study of superoscillations initiated by Aharonov et al.,
who first raised the question about extracting a particle from a superposition state with an energy
greater than that of its highest mode [1–11]. Transient zeros of the wavefunction have also been
considered in the study of quantum revival and quantum carpets [12–14].
We consider only the 1-D infinite square well, but the findings here can be trivially generalized
to the 3-D case. We proceed with the simplest example of the effect in question, and after this, we give
the general derivation for arbitrary superposition states.
The measurement of an alternate energy eigenbasis is performed in two stages. In the first stage,
at a moment when there is a zero in the wavefunction, an infinite delta-function potential barrier
Entropy 2016, 18, 149; doi:10.3390/e18040149
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is suddenly raised at the location of a zero, which has the effect of dividing the original infinite
square well into two adjacent infinite square wells, while causing virtually no perturbation to the
wavefunction (a similar process is discussed in [15], although with quite a different purpose, and an
analysis of perturbation theory with singular potentials is given in [16]). This division results in a
superposition state of the particle being on one side of the barrier or the other and, furthermore, a
superposition of the energy levels of each individual well. We call the combined spectra of the two
individual wells an interference spectrum. This process has effectively accomplished a spectrum and,
thus, frequency conversion of the state, which may be quite novel when compared to other related
techniques [17–24].
In the second stage, the energy of the state is measured and is now found in an energy eigenstate
of one of the two new wells, rather than an eigenstate of the original well. This is the real effect of
raising the barrier: it changes the list of eigenstates onto which the state can collapse when measured.
Of particular interest is the fact that in the new spectral decomposition of the state, it may be
possible to measure an energy higher than the energy of the highest mode in the original spectral
decomposition of the state. In general, there is no evidence of a violation of conservation of energy
because the sudden barrier introduces a large energy uncertainty due to the energy-time uncertainty
relation [25–29].
A wavefunction that contains a region of superoscillation turns out to be a special case of this
phenomenon, wherein very particular superposition states have transient zeros that remain stable
for extended durations [4]. Because of the stability of these zeros, barriers can be raised very slowly,
and the new spectrum can be obtained without introducing a large energy uncertainty, which may be
interpreted as causing a violation of the conservation of energy [1].
Here, we propose that all we have done through this two-stage process is to effect a measurement
in an energy eigenbasis that does not commute with the original energy eigenbasis of the state.
The barrier can be raised with virtually no perturbation to the wavefunction, and this has the
effect of changing the discrete energy spectrum of that wavefunction. With this interpretation, the
wavefunction itself does not have a definite preferred energy spectrum until it is measured with
specific boundary conditions (i.e., one spectrum without the barrier or another with the barrier).
This nullifies the issue of a violation of the conservation of energy, since the original spectrum places
no special restriction on what energies can be obtained from a measurement.
While the idea of alternative energy eigenbases for the infinite square well may seem somewhat
radical, we point out that a spin- 12 particle can be measured in complementary Pauli-spin eigenbases,
and each measurement is performed by coupling the spin to a different Hamiltonian. Just as in the
present case, the change of spectrum has no effect on the state; all that has changed is the list of
eigenstates into which the particle can collapse when a measurement of the energy is performed.
Finally, we have performed extensive numerical simulations of the evolution of several key
wavefunctions as a Gaussian barrier of various widths is raised at various rapid speeds to a finite
potential. We used the simulation data over this range of parameters to come up with an approximate
characterization of how a narrow barrier changes the wavefunction as a function of the barrier’s
speed, width and the characteristics of the initial state.
Our simulations verify that if a very narrow barrier is raised sufficiently fast at a zero of
the wavefunction, the splitting of the well and the change of energy spectrum can indeed be
accomplished with virtually no change to the wavefunction.
The remainder of this paper is organized as follows: in the next section, we explore the simplest
case of a wavefunction with a single transient zero in complete detail and introduce an example case
that might allow experimental verification of these ideas. Next, we discuss the details of raising the
barrier; the sudden and adiabatic approximations and the parametric conversion of the spectrum
and splitting of the the eigenstates. After this, we discuss a preliminary idea for an experimental
implementation of this effect. We then present the formalism for the general interference spectrum of
a general superposition state. Finally, we discuss energy conservation and energy-time uncertainty
Entropy 2016, 18, 149
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in alternative interpretations of this effect and end the paper with a few concluding remarks. In the
Appendix, we discuss the numerical simulations of the time-dependent Schrödinger equation that
we conducted in order to characterize the effect of rapidly raising a Gaussian barrier.
2. Results
2.1. The Simple Case
To begin, we will take our infinite square well, which we will call Well 0, to be of width L, with
boundaries located at x = 0 and x = L. The energy eigenstates of this well are,
√
ψl0 (x)
=
2
lπx
sin
,
L
L
(1)
h̄2 π 2 l 2
.
2ML2
(2)
and have corresponding energies,
E0 (l) =
Consider the following normalized superposition of the ground state (l = 1) and first excited
state (l = 2),
√
2
πx
2πx
ψ(x) =
(α sin
− sin
),
(3)
L(α2 + 1)
L
L
with α ≡ 2 cos [ πxL 0 ], and x0 ∈ (0, L2 ) is a zero of ψ(x). This zero is transient and quickly vanishes as
the state evolves in time. During a complete period of evolution, this state also develops a transient
zero at x1 = L − x0 , and so, we define the list of zero points for Ψ(x, t) as s = {(x0 , t0 ), (x1 , t1 )}.
Thus, at any given time, this function has at most one zero inside the well, and by symmetry, we
only need to consider the case of (x0 , t0 ). This zero is technically only present at a single instant in
time, and thus, the barrier must be raised instantaneously. Clearly, both the delta-function potential
and the sudden implementation are the nonphysical ideal case.
Now, suppose that at time t0 = 0, we raise a new infinite delta-function potential barrier at
x0 , splitting the original well into two smaller wells of widths x0 and L − x0 , which we will call
Well 1 and Well 2, respectively. ψ(x) already satisfies the new boundary conditions, and so, there
is no instantaneous change in the wavefunction or the expectation value of any observable. The
probabilities to find the particle in either well are,
P1 = ∫
x0
0
L
∣ψ(x)∣2 dx,
P2 = ∫
x0
∣ψ(x)∣2 dx.
(4)
Defining the truncated and renormalized wavefunctions in each well as:
ψ1 (x) = {
√
ψ(x)/ P1
0
0 ≤ x ≤ x0
x0 < x ≤ L
ψ2 (x) = {
0
√
ψ(x)/ P2
0 ≤ x ≤ x0
,
x0 < x ≤ L
and
we can rewrite the original wavefunction as:
√
√
ψ(x) =
P1 ψ1 (x) +
P2 ψ2 (x).
(5)
If we note that after the barrier goes up, a classical particle must either be in Well 1 or Well 2,
we can interpret this wavefunction as a superposition of the particle being in Well 1 in the state ψ1 (x)
with probability P1 or in Well 2 in the state ψ2 (x) with probability P2 .
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Defining,
L
⟨E⟩ = ∫
0
ψ∗ (x) Ĥψ(x)dx,
(6)
ψ1∗ (x) Ĥψ1 (x)dx,
(7)
ψ2∗ (x) Ĥψ2 (x)dx,
(8)
x0
⟨E1 ⟩ = ∫
0
and:
L
⟨E2 ⟩ = ∫
x0
gives us the relation,
⟨E⟩ = P1 ⟨E1 ⟩ + P2 ⟨E2 ⟩.
(9)
The state has expectation value ⟨E⟩ because with probability P1 , the particle is in Well 1 with
average energy ⟨E1 ⟩, and with probability P2 , it is in Well 2 with average energy ⟨E2 ⟩.
Wells 1 and 2 have energy eigenstates,
ψn1 (x)
and:
√
⎧
nπx
2
⎪
⎪
x0 sin x0
=⎨
⎪
⎪
⎩ 0
⎫
0 ≤ x ≤ x0 ⎪
⎪
⎬,
⎪
x0 < x ≤ L ⎪
⎭
⎧
⎪
⎪ 0
2
ψm
(x) = ⎨ √ 2
mπ(x−x0 )
⎪
⎪
L−x0 sin
L−x0
⎩
⎪
0 ≤ x ≤ x0 ⎫
⎪
⎬,
⎪
x0 < x ≤ L ⎪
⎭
respectively, with corresponding energy eigenvalues,
h̄2 π 2 n2
,
2Mx02
(10)
h̄2 π 2 m2
.
2M(L − x0 )2
(11)
E1 (n) =
and:
E2 (m) =
In general, these energy levels are different from one another and from E0 (l). Furthermore, it is
possible to measure an energy E1 (n) or E2 (m) larger than E0 (2), which is the highest mode of Well 0
present in the superposition state ψ(x). This is then an example of a superoscillatory effect.
If we measure the energy of the original well, we will find energy E0 (1) with probability
2
α /(α2 + 1) and energy E0 (2) with probability 1/(α2 + 1), indicating that we have projected the state
onto states ψ10 (x) or ψ20 (x). If instead, we split the well by putting up the barrier at x0 and then
2
measure the energy, the measurement projects onto states ψn1 (x) or ψm
(x).
To find the probability to collapse onto eigenstates of the split well, we decompose ψ(x) into the
modes of the split wells:
∞
∞
n=1
m=1
2
ψ(x) = ∑ an ψn1 (x) + ∑ bm ψm
(x)
(12)
with an and bm as shown below:
an = √
=
bm = √
2
x0
L(α2 + 1)
∫0
x0
(α sin
πx
2πx
nπx
− sin
) sin
dx
L
L
x0
3√
πx
0
sin 2πx
⎞
2nL 2 x0 (−1)n ⎛ α sin L 0
L
√
−
2
2
2
2
2
2
4x0 − n L ⎠
⎝ x0 − n L
π α2 + 1
2
L
πx
2πx
∫ (α sin L − sin L ) sin
(L − x0 )L(α2 + 1) x0
mπ(x − x0 )
dx
L − x0
(13)
(14)
(15)
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3
−2mL 2
=
π
√
0
α sin πxL 0
sin 2πx
⎞
L − x0 ⎛
L
−
.
2
2
2
2
2
2
2
α + 1 ⎝ (L − x0 ) − m L
4(L − x0 ) − m L ⎠
(16)
The mod-squared coefficients ∣an ∣2 and ∣bm ∣2 are then the probabilities to find the particle in each
eigenstate. Additionally, of course,
∞
∞
n=1
m=1
2
2
∑ ∣an ∣ + ∑ ∣bm ∣ = P1 + P2 = 1.
(17)
2
The two alternate energy eigenbases ({ψl0 (x)} and {ψn1 (x), ψm
(x)}) each span the space of
normalizable functions that are zero at x0 , but the corresponding Hamiltonians do not commute;
thus, these two energy eigenbases are complementary (or at least, there is some uncertainty relation
between them).
We call the new energy spectrum that is available using this measurement procedure the
interference energy spectrum of the state ψ(x). This name is appropriate because the available energies
that can be measured are only different from E0 (l) if ψ(x) is a superposition of different ψl0 (x).
The simplest way to see this is by considering the state ψ20 (x) by itself, which is also obtained by
considering the above treatment for ψ(x) in the limit that x0 → L/2. This state has a definite energy
2
2
π
and a stationary zero at x = L/2. If we insert an infinite barrier at this zero
⟨E⟩ = E0 (2) = 2h̄
ML2
and split the well, we get equal probability to find the particle in the ground state of each sub-well
2
2
π
= E0 (2). Thus, even if we split the well, we always
(left or right), with energy ⟨E1 ⟩ = ⟨E2 ⟩ = 2h̄
ML2
find the particle with the same wavelength and, thus, the same energy. It is trivial to see that this
generalizes to splitting any eigenstate at any subset of its nodes.
Of particular interest for experiments would be to prepare the state ψ(x) with x0 = 38 L, shown in
Figure 1. If we put up the barrier at x0 and measure the energy, we find that the probability to find the
particle in the ground state of the first well, P(n = 1) ≈ 6%, with energy E1 (1) =
2
2
64 h̄2 π 2
9 2ML2 , which exceeds
h̄ π
16
the energy E0 (2) = 4 2ML
2 of the highest mode in ψ(x) by a factor of 9 . Thus, if the experiment can
be performed, it should be possible to measure superoscillatory interference energies with plausible
success rates.
2
*(x)
1
0
-1
L/4 x0 L/2
3L/4
L
Figure 1. Plot of ψ(x) with x0 = 83 L where a sudden barrier can divide the well.
For all values of x0 , the ground states of either split well are always the most probable, with the
probabilities, ∣an ∣2 and ∣bm ∣2 , of measuring higher modes converging toward zero as n, m → ∞. It is
nevertheless possible to measure arbitrarily high energy outcomes with nonzero probability, whereas
only the two lowest modes were present in the original well.
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2.2. Raising the Barrier: The Sudden Approximation and the Adiabatic Limit
We made the assumption above that if the delta-function barrier is raised very quickly at a
transient zero of the wavefunction, then the wavefunction itself is not significantly changed by the
process, and as a result, the expectation values of all observables are also unchanged. Because both
a delta-function barrier and an instantaneous potential change are nonphysical, we have performed
extensive numerical simulations of the time-dependent Schrödinger equation with a Gaussian barrier
of varying widths w, raised linearly to a large finite height over a finite period of time that is very
short relative to the characteristic frequencies of the initial states. The simulation was run over a
representative range of physically plausible parameters, with emphasis on the narrow-barrier regime.
We used ψ(x) with x0 = 38 L for the simulation, which was performed using a modified
fourth-order Runge–Kutta method. More technical details about the simulation can be found
in Section 5.
The results of the simulation show that as the barrier is made wider, the change to the energy
of the state grows smaller and goes to zero, and in the nonphysical limit that the width goes to zero
(the delta-function limit), it appears to go to infinity as 1/w3 (see Equations (45) and (46). This can
be overcome in the equally nonphysical limit that the barrier is raised instantaneously, in which case
there is no change in energy. We do find that for physically-reasonable barrier widths, final barrier
magnitudes and raising periods, the lowest modes of the well can be effectively split with a negligibly
small perturbation to the state and its energy. We obviously do not get the exact spectrum we would
if the well had been split by a delta-function, but the spectrum and eigenstates are certainly close
enough to obtain a superoscillatory energy.
For example, set the width to w = 10−3 [L] and the total period to raise the barrier to a maximum
scale of V = 104 [h̄2 /2ML2 ] to τ = 10−10 [2ML2 /h̄]. The kinetic energy of the original state is
⟨K⟩ = 2.8918[h̄2 π 2 /2ML2 ], and the change in kinetic energy is on the order of 10−9 [h̄2 /2ML2 ], which
is below the error threshold of the simulation. With this barrier, the ground state kinetic energy is
E(1) ≈ 2.5605[h̄2 π 2 /2ML2 ] (this is a numerical result), compared to E(1) = 2.5600[h̄2 π 2 /2ML2 ] for
the delta-function barrier, and the corresponding wavefunctions are also nearly identical; thus, the
desired splitting has been performed.
We have also considered the adiabatic limit in which a delta-function barrier at x0 = 38 L is raised
very slowly, such that energy levels of the original un-split well transition gradually into the energy
levels of the two wells after splitting. The shifting of the first seventeen energy levels is shown in
√
Figure 2 in terms of the wave number k = 2ME. This figure and also Figures 3 and 4 were obtained
by parametric solutions of the time-independent Schrödinger equation, rather than a simulation of
adiabatic evolution in time.
In general, as the barrier magnitude V increases, many eigenstates of the well become gradually
more and more confined to one side of the barrier or the other, with tunneling rates vanishing as
the barrier becomes infinite. Figure 3 shows the ground state and first excited state of the well as a
function of the potential of the delta-function barrie.
However, this is not true of all eigenstates, which leads to some interesting physics for the cases
where the two new wells have degenerate energy levels. In these cases, the adiabatic approximation
fails, strictly speaking, because there will be a significant probability of transitions between the
nearly-degenerate levels.
Entropy 2016, 18, 149
k
L
!: "
l=17
l=16
l=15
l=14
l=13
l=12
l=11
l=10
l=9
l=8
l=7
l=6
l=5
l=4
l=3
l=2
l=1
7 of 20
88/5
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
m=11
16 m=10, n=6
72/5
m=9
40/3
64/5
n=5
m=8
56/5
32/3
m=7
n=4
48/5
m=6
8
10
100
m=5, n=3
32/5
m=4
16/3
24/5
n=2
m=3
16/5
8/3
m=2
n=1
8/5
m=1
1000
2
V ( 2M7h L2 )
*(x)
Figure 2. Spectrum of the infinite square well with a delta-function potential of magnitude V located
√
at x0 = 38 L. The levels are shown in terms of the wave number k = 2ME and a logarithmic scale for
V (which has units of [h̄2 /2ML2 ]).
V = 0
2
V = 8
2
1
x0
L
0
1
0
0
L
V = 8
V = 0
*(x)
x0
x0
L
0
0
V = 20
x0
L
0
2
2
2
2
1
1
1
1
1
0
0
0
0
0
-1
-1
-1
-1
-1
-2
-2
0
x0
L
0
x0
L
-2
0
x0
L
x0
L
V = 1000
V = 40
2
-2
V = 1000
2
1
0
0
0
V = 40
2
1
1
0
V = 20
2
-2
0
x0
L
0
x0
Figure 3. The ground state (top) and first excited state (bottom) of the infinite square well with a
delta-function potential of magnitude V located at x0 = 83 L. Clearly, as the barrier magnitude increases
each eigenstate becomes confined on one side of the barrier or the other, becoming eigenstates of the
individual wells.
L
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*(x)
2
8 of 20
2
V = 0
0
-2
0
*(x)
2
x0
V = 40
0
0
-2
-2
L
0
2
V = 0
2
x0
2
2
V = 150
0
2
x0
L
V = 80
-2
0
2
x0
L
0
2
V = 150
0
0
0
0
0
-2
-2
-2
-2
-2
0
x0
L
0
x0
L
0
x0
L
V = 1000
0
-2
0
L
V = 40
V = 80
0
x0
L
x0
L
V = 1000
0
x0
L
Figure 4. The seventh (top) and eighth (bottom) excited state of the infinite square well with a
delta-function potential of magnitude V located at x0 = 83 L. These eigenstates become degenerate
and clearly fail to become confined on one side of the barrier or the other as the barrier magnitude
increases, and thus they do not become eigenstates of the individual wells. Instead, the eigenstates
of the individual wells are orthogonal superpositions of these two eigenstates of the original well, as
shown in Equations (18) and (19).
The l = 8 energy level of the unsplit well is degenerate with the n = 3 and m = 5 levels of the
two wells after splitting (E0 (8) = E1 (3) = E2 (5)), but a single energy level cannot divide into two
orthogonal energy levels under adiabatic evolution. In fact, it is easy to see analytically that because
the l = 8 mode has a node at x0 , it will remain unchanged no matter how quickly or slowly the barrier
is raised, meaning that it does not become confined to one side of the barrier, but rather becomes a
superposition of the particle being on either side. As the barrier goes up, the l = 7 mode gradually
becomes degenerate with the l = 8 mode, but also fails to become confined to one side of the barrier
(see Figure 4). The l = 7 mode develops a slope-discontinuity at x0 , such that the l = 8 and l = 7
eigenstates remain orthogonal, even as their energies become degenerate,
ψ80 (x)
lim
V→∞
√ ⎧
8πx
2⎪
⎪ sin L
⎨
=
8π(L−x)
L⎪
⎪
L
⎩ − sin
ψ70 (x)
⎧
8πx
⎪
⎪ sin L
= A⎨ 3
8π(L−x)
⎪
⎪
L
⎩ 5 sin
0 ≤ x ≤ x0
x0 < x ≤ L
0 ≤ x ≤ x0
x0 < x ≤ L
where A is a normalization constant.
Remarkably, since neither state becomes confined, it is not the case that the l = 7 and l = 8
eigenstates of the original well gradually become the n = 3 and m = 5 eigenstates of the two wells
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after splitting. Instead, the confined eigenstates of the two wells are superpositions of the these two
degenerate states of the unsplit well. In the limit of infinite V, the left well n = 3 eigenstate is,
√
ψ31 (x)
⎛
3A
= B ψ70 (x) +
5
⎝
L 0 ⎞
ψ (x) ,
2 8
⎠
(18)
and the right well m = 5 eigenstate is,
√
ψ52 (x)
⎛
= C ψ70 (x) − A
⎝
L 0 ⎞
ψ (x) ,
2 8
⎠
(19)
where B and C are new normalization constants.
For x0 = 38 L, the same thing happens for all pairs of levels l = 8s and l = 8s − 1 for all integers
s ≥ 1 and with the same coefficients in the superposition. In general, this effect happens whenever the
specified degeneracy condition is present for any location of x0 where a barrier is raised.
2.3. Proposed Experiment
The analysis of this paper can be applied equally well to a photon in a cavity, and this leads to a
proposition for a simple experimental test of the ideas we present here, which would ultimately take
the form of a type of frequency converter, similar to other work using superoscillations [17].
The idea is to use a square multimode fiber that acts as an infinite square well in two dimensions
while being effectively free in the third dimension. If the superposition state of Equation (3) can be
created in one or both of the constrained dimensions, then there would be particular positions along
the free dimension, corresponding to specific propagation times, for which the zero would be present
in the wavefunction. A split in the fiber could begin at the location of such a zero, with the split now
playing the role of a barrier that is raised very quickly inside the infinite square well, with the effective
quickness coming from the propagation speed of the photon along the longitudinal direction.
Figure 5 shows the time evolution ∣ψ(x, t)∣2 of the initial state ψ(x) of Equation (3), such that
the state evolves within the original well for one revival period; then, the infinite barrier appears
suddenly at x0 = 38 L, and the state evolves for the same period in the new potential. The presence of
higher energy modes with small amplitudes is clearly visible in the erratic behavior of ∣ψ(x, t)∣2 after
the barrier is raised.
If our analysis is correct, then this should result in frequency conversion in the transverse
mode(s) of the photon, such that the spectral superposition of a photon that emerges after the
split will be changed, although the average energy is nearly unchanged. Components of the new
superposition that remain inside the operating band of the multimode fiber will be incident on
transverse frequency-sensitive detectors along the line, and components that are outside the band
may escape, but could also potentially be captured by external detectors.
We should again stress that this frequency conversion is not in the direction of propagation along
the fiber, but rather, it is the modes oscillating perpendicular to this direction that are converted.
Numerous technical details will need to be addressed before this experiment can be realized, but
this goes beyond the scope of this paper.
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x
t
Figure 5. The initial probability amplitudes for ∣ψ(x, 0)∣2 appear on the right of the figure, and time
evolves to the left. After one revival period, when the zero has reappeared at x0 = 38 L, an infinite
barrier is raised there, and the state evolves for the same period in the new potential.
2.4. The Interference Energy Spectrum
In this section, we develop the general theory of interference spectra for arbitrary superposition
states. True superoscillatory functions are a special case of this phenomenon that contain stable zeros,
which allow barriers to be raised much more slowly; although it remains unclear if they can be raised
slowly enough to justify the adiabatic approximation.
Let us begin with a general superposition state of the 1-D square well,
∞
Ψ(x, t) = ∑ cl ψl0 (x)e
−iEl t
h̄
.
(20)
l=1
This state has a unique set of zero points S = {(xi , ti )}, such that Ψ(xi , ti ) = 0 and 0 < xi < L for
all (xi , ti ) ∈ S. At a given time τ, it is possible to place infinite barriers at any or all points (xi , ti ) for
which ti = τ in order to split the well into smaller wells.
Let us suppose that at time τ, we choose to divide the well into N + 1 smaller wells by placing
barriers at a set of N available zeros from S, {χ j } with j = 1, ..., N. We append the two endpoints χ0 = 0
and χ N+1 = L to this set and expand the index, such that χ j > χ j−1 is defined for all j = 1, ..., N + 1. The
resulting interference energy spectrum that is now available is,
Ej (k j ) =
h̄2 π 2 k2j
2M(χ j − χ j−1 )2
,
(21)
for wells j = 1, ..., N + 1, where k j = 1, 2, ..., ∞ is the quantum number of the j-th well.
In general, the union set of all possible spectra that can be obtained by placing any number of
barriers at any number of points (xi , ti ) in S is the complete interference spectrum of the state Ψ(x, t).
We now switch to the Hilbert space representation of this problem in order to give the calculated
probabilities, noting that the domain of each well, (χ j−1 , χ j ), is a different Hilbert space. We introduce
the ket ∣0⟩ to represent the original well and a d-level quantum system with basis {∣j⟩} to represent
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the different split wells, with d = N + 1. Each well is an infinite-dimensional Hilbert space, but the
projectors onto each well have relative rank (subscript) proportional to the well size, such that,
N+1
∣0⟩⟨0∣ L ≡ ∑ ∣j⟩⟨j∣χ j −χ j−1 .
(22)
j=1
To begin, we represent Ψ(x, τ) as,
∞
∞
N+1
∣Ψ⟩ = ∑ dl ∣l⟩∣0⟩ = ∑ dl ∑ ∣l j ⟩∣j⟩,
l=1
l=1
(23)
j=1
where ∣l⟩ represents ψl0 (x), and ∣l j ⟩ is the unit ket for the (unnormalized) truncated eigenfunction in
the j-th well, such that {∣j⟩∣l j ⟩} is a complete orthonormal basis.
We compute the similarity matrices { Â j } that perform the change of basis on each well,
∞ ∞
 j = ∣j⟩⟨j∣ ∑ ∑ Ak j l ∣k j ⟩⟨l j ∣,
(24)
k j =1 l=1
where ∣k j ⟩ are the normalized energy eigenstates of the j-th well, corresponding to,
j
ψk (x)
j
¿
Á
À
=Á
k j π(x − χ j−1 )
2
sin [
],
χ j − χ j−1
χ j − χ j−1
(25)
and the matrix elements are given by,
Ak j l = ⟨k j ∣l j ⟩ = ∫
2
χj
√
χ j−1
sin [
L(χ j − χ j−1 )
k j π(x − χ j−1 )
χ j − χ j−1
] sin [
lπx
] dx
L
⎤
⎡
⎢ (−1)k j sin ( lπχ j ) − sin ( lπχ j−1 ) ⎥
⎥
⎢
2k j L √
⎥
⎢
L
L
⎥.
=
L(χ j − χ j−1 ) ⎢⎢
⎥
2
2
2
2
π
l (χ j − χ j−1 ) − k j L
⎥
⎢
⎥
⎢
⎥
⎢
⎣
⎦
(26)
(27)
Finally, letting the matrices  j act on the wavefunction, we obtain the representation in terms of
the new energy eigenbasis:
⎡∞ ∞
⎤ ∞
N+1
N+1
⎢
⎥
′
′
∑ Â j ∣Ψ⟩ = ∑ ∣j⟩⟨j∣ ⎢⎢ ∑ ∑ Ak j l ∣k j ⟩⟨l j ∣⎥⎥ ∑ dl ′ ∑ ∣l j′ ⟩∣j ⟩
⎢
⎥
′
′
j=1
j=1
j =1
⎣k j =1 l=1
⎦ l =1
N+1
N+1
∞ ∞
= ∑ ∣j⟩ ∑ ∑ dl Ak j l ∣k j ⟩.
j=1
(28)
(29)
k j =1 l=1
In this form, it is clear that this is an entangled state in the sense that the energy levels are
correlated to specific wells. If we project onto a particular well ∣j⟩, we get a superposition of the
energy eigenstates of that well. Likewise, if we project onto a specific energy eigenstate ∣k j ⟩, we are
also projecting onto a specific well (or several wells with degenerate energies).
The probability of finding the particle in the j-th well with energy Ek j is,
RRR ∞
RRR2
P(k j ) = RRRRR∑ dl Ak j l RRRRR ,
RRRl=1
RRR
(30)
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and the total probability of finding it in each well is,
P(j) = ∫
RRR2
∞ RRR ∞
∣Ψ(x)∣2 dx = ∑ RRRRR∑ dl Ak j l RRRRR .
χ j−1
RRR
k j =1 RRRl=1
χj
(31)
3. Discussion
The results given here raise interesting questions about the conservation of energy. Unlike the
genuine superoscillating functions that have been studied elsewhere, the method we are using to
split the wells requires that the barrier be raised very quickly, and because the ∆t of this process is so
small, it necessarily introduces a large ∆E due to the Heisenberg uncertainty principle. Thus, even
though ⟨E⟩ is unchanged by the sudden addition of the barrier at a zero of the wavefunction, the
change in energy between the individual pre-barrier and post-barrier levels can be supplied by the
∆E. Therefore, in this case, the uncertainty principle washes out any possibility that the conservation
of energy is violated.
In general, one may consider the condition ∆⟨E⟩ = 0 sufficient to show that energy conservation
is obeyed. Even though the Hamiltonians are different before and after the barrier is present, the
complete eigenbases of the two Hamiltonians both span the space of normalizable functions on the
interval x ∈ [0, L] with zeros at x = {0, x0 , L}. If ∣ψ⟩ is a true superposition state, then there is nothing
unexpected about finding energies E1 (n) or E2 (m) when the energy is measured in the eigenbasis
{∣n⟩, ∣m⟩}. The potential problem arises if one assumes that the discrete spectrum E0 (l) is an inherent
property of the state ∣Ψ⟩, and it should be impossible to measure other energies.
Supposing there is an inherent preferred spectrum, then as the barrier goes up, the energy levels
of the original well divide and smoothly transition to the energy levels of the split well. The particle
only interacts with the barrier, and so, by energy conservation, it must be the case that the change in
energy is supplied by the barrier. If the particle is found with energy eigenvalue k j in the split well,
it has probability Pk j (l) to have transitioned from energy eigenvalue l of the original well, for which
the barrier must have supplied the energy change ∆Ek j l = Ej (k j ) − E0 (l). We thus define the barrier
state ∣Bk j l ⟩ as the state in which the barrier lost this energy. Each split-well energy eigenstate of the
joint particle-barrier system PB is then of the form,
∞
∣k j ⟩⟨k j ∣PB = ∑ Pk j (l)∣l j ⟩⟨l j ∣∣Bk j l ⟩⟨Bk j l ∣.
(32)
l=1
Thus, we see that after the barrier is raised, the state ∣Ψ⟩ is entangled with the barrier through
energy conservation. The amount of free energy needed to produce the individual shifts ∆Ek j l is
easily provided by the large uncertainty ∆E for a sudden barrier.
It should be possible to experimentally test this interaction by performing an ensemble of
runs of the experiment and taking the average of all barrier-energy measurements conditioned
on post-selecting a particular energy eigenstate ∣k j ⟩ of the particle in the split well. This average
should wash out the noise introduced by the large ∆E, making it possible to measure the average
barrier energy,
∞
⟨E B ⟩ = − ∑ Pk j (l)∆Ek j l .
(33)
l=1
Now, there are several viable interpretations of how Pk j (l) should be defined if we wish to
assume a preferred spectrum ∣l⟩ as an inherent property of the superposition state: a property not
specified by the wavefunction alone.
One simple choice is Pk j (l) = ∣dl ∣2 , which means that regardless of which ∣k j ⟩ the state is found
in, the probability that it transitioned from energy level E0 (l) is the same as the probability to find
the state with that energy in the original well. Using this form for our simple example case with
x0 = 3L/8 and the initial superposition state of Equation (3), the probability to find the particle in the
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ground state of the smaller well, with energy (8/3)2 E0 (1), is roughly 6%, and the average energy of
the barrier, post-selected on this outcome, is ⟨E B ⟩ = −4.22E0 (1). This case is particularly interesting
because the wavefunction is zero at the location of the barrier, and yet, the particle and barrier seem
to exchange significant quantities of energy as the barrier is raised.
Alternatively, we can consider a quasi-probability treatment of the superposition state. Given
the initial state ∣Ψ⟩ corresponding to Equation (3) and the final state ∣k j ⟩, the best estimate for the
probability that the intermediate state was ∣l⟩ is given by the weak value of the projector ∣l⟩⟨l∣ [30–33],
P̃k j (l) = R
⟨k j ∣l⟩⟨l∣Ψ⟩
=R
⟨k j ∣Ψ⟩
Ak j l dl
∞
∑l ′ =1
Ak j l ′ dl ′
.
(34)
We should stress that this quasi-probability can be less than zero or greater than one, which
makes its physical interpretation somewhat unclear. Indeed, in our example case, post-selecting on
the ground state of the smaller well, we find that the quasi-probabilities that the previous energy
level was l = 1 or l = 2 are P̃k1 (1) = −1.037 and P̃k1 (2) = 2.037, respectively. Nevertheless, if we
use ∣k j ⟩⟨k j ∣ = ∑∞
l=1 P̃k j (l)∣l j ⟩⟨l j ∣∣Bk j l ⟩⟨Bk j l ∣ after the barrier is raised, then the average energy of the
barrier, post-selected on this outcome, is ⟨E B ⟩ = 0, which is quite a striking result, that turns out
to be completely general, as we now show.
√
2
lπx
For a general superposition state Ψ(x) = ∑∞
d
l
l=1
L sin L , the condition that Ψ(x) has a zero at
lπx0
x0 is simply ∑∞
l=1 dl sin L = 0. If a barrier is placed at x0 for a general wavefunction and the particle
subsequently collapses into eigenstate ∣k1 ⟩ of the well between x = 0 and x = x0 , the average energy of
the barrier using the quasi-probability, P̃k1 (l), is,
∞
lπx0
L
′
0
sin l πx
L
∑l=1 dl sin
⟨EkB1 ⟩ =
dl ′
∆Ek l ′
∞
∑l ′ =1
1
.
(35)
Thus, we see that for any initial wavefunction with a zero at x0 , the average energy of a barrier
that is suddenly raised at x0 , conditioned on the post-selection of any one outcome ∣k1 ⟩ in the split
well, is always zero. We used a quasi-probability distribution to obtain this result, but we are not
actually predicting that any physical event occurs with probability P̃k1 (l); rather, it is used as an
intermediate calculation tool to address the fact that the initial state was a superposition of multiple
∣l⟩ eigenstates. Quasi-probabilities outside the range zero to one are also known to be related to
quantum contextuality [34,35].
There is one other definition one might use for Pk j (l), which are the probabilities for the mixed
2
state prepared as ρ = ∑∞
l=1 ∣dl ∣ ∣l⟩⟨l∣. This is the least physical choice, because the quantum interference
terms have been removed, and the predicted probabilities of different outcomes are entirely different
than for ∣Ψ⟩. Nevertheless, ρ does explicitly have the energy spectrum ∣l⟩⟨l∣, which is unclear for ∣Ψ⟩,
and thus, it may be relevant here. We can obtain,
Pk j (l) =
∣Ak j l ∣2 ∣dl ∣2
∑l ′ =1 ∣Ak j l ′ ∣2 ∣dl ′ ∣2
∞
,
(36)
and using these probabilities in our example case, we obtain ⟨E B ⟩ = −3.73E0 (1). In this case, the
exchange of energy between the particle and barrier can be explained by local interaction, since the
individual ∣l⟩ are not generally zero at x0 .
Regardless of which explanation we use for the shifts of individual levels, the wavefunction is
unchanged, and the overall average energy provided by the barrier is still zero. This raises interesting
questions about the physics of measuring any quantum state in an alternate eigenbases, even a spin.
If the preferred spectrum (both eigenvalues and eigenstates) is an inherent property of the quantum
state of a system, then changing the basis prior to a measurement has a direct effect on that inherent
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property, but without changing the state vector itself (i.e., without collapse). This would then be an
explicit manifestation of quantum contextuality [36,37], in that the choice of measurement context
physically changes an internal property of the state: the preferred basis.
Conducting the experiment to measure the barrier energy may prove technically challenging,
but it would allow us to test our suppositions for the form of Pk j (l).
On the other hand, taking the viewpoint that the state has no inherent discrete energy spectrum
(and thus, Pk j (l) is meaningless) and noting that the barrier cannot interact locally with the particle,
which has zero probability to be found in the same place as the barrier, then the average energy of the
barrier should be ⟨E B ⟩ = 0, regardless of post-selection.
It is interesting that even this experiment cannot distinguish the case of an inherent preferred
spectrum with a quasi-probability distribution from the case of no inherent spectrum at all, since
both predict ⟨E B ⟩ = 0.
Finally, if we consider the case of a superoscillating wavefunction [1], the barrier can be raised
very slowly at quasi-stable nodes of the superoscillation, and the large ∆t leads to a small ∆E, small
enough that it may not be enough to encompass the individual shifts ∆Ek j l . Thus, if we insist that
the wavefunction has an inherent preferred spectrum, we may be presented with a violation of
conservation of energy, especially if ⟨E B ⟩ ≠ 0. If ⟨E B ⟩ = 0, then it may still be that the barrier simply
facilitates exchanges of energy between different levels of the particle, in which case its own ∆E may
not matter. On the other hand, if the wavefunction has no inherent preferred spectrum, there are no
shifts ∆Ek j l , and the issue vanishes.
4. Materials and Methods
The details of our numerical simulation of the time-dependent Schrödinger equation can be
found in the Appendix. The simulation was written in MATLAB 2014a, and the code and data are
available upon request.
5. Conclusions
We have explored the idea, and verified through simulation, that the energy eigenbasis of a state
of the infinite square well can be altered through the addition of sudden potential barriers without a
change to the state or its average energy. We consider the interpretation that this is a measurement
in an alternate energy eigenbasis, because the energy eigenstates onto which the particle can collapse
after the barrier is raised are different than the eigenstates of the original well and have different
energies. The main point of contention with this view is that it is common to interpret a superposition
state of the infinite square well as having a discrete list of preferred spectral energies as an inherent
property, such that it is impossible to measure any energy not on this list. This is inconsistent with
the idea that a genuine superposition state can be measured in different bases and can collapse onto
any eigenstate in the measured basis. If we ascribe an inherent discrete spectrum directly to the
superposition state, then the barrier must exchange energy with the particle in order to adjust that
spectrum. This interaction must occur despite the fact that the particle has zero probability to be
found at the location where the barrier is raised (∣ψ(x0 )∣2 = 0), but this may be consistent, because
the discrete energy spectrum of a bound particle is not usually considered a local property of the
wavefunction; thus, the inherent spectrum must be a de-localized, or possibly nonlocal, property
of the quantum state of the particle. In principle, an experiment might allow us to determine if
this inherent preferred spectrum exists by measuring the barrier energies post-selected on obtaining
particular measurement outcomes.
The alternative viewpoint, that the state has no inherent discrete spectrum, raises interesting
questions about dynamical collapse, since it implies that the particle must somehow probe the entire
shape of its binding potential as part of the dynamical collapse process, in order to cause a collapse
into an eigenstate of the correct Hamiltonian.
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Following this research, we have continued to explore the idea that there is no preferred
discrete energy spectrum inherent to a wavefunction at all; but rather, it is always the measurement
Hamiltonian that determines the spectrum, and this is where quantization appears. The wavefunction
itself is not quantized, and its evolution can be modeled by considering its Fourier transform into a
continuous spectrum of plane waves. While this work is ongoing, one preliminary result of some
interest is the fact that for certain states, there are discrete zeros in the Fourier transform of the
state, which means that in any discrete energy eigenbasis that can be used to measure the state,
the probability of obtaining that spectral energy is zero. Thus, while the allowed energy levels of
such wavefunctions are not generally quantized, the forbidden energies are quantized. We call this
amusing phenomenon unquantum mechanics. The only states that do have discrete quantum spectra
in the Fourier transform domain are unnormalizable continuous plane waves. We plan to develop
these ideas further in a subsequent paper.
We have found little in the literature that seems specifically relevant to the new ideas presented
here, so we provide a collection of citations on work that is somewhat more distantly related in order
to flesh out the state of the art. These topics include, frequency conversion using nonlinear optics
and other systems, double-well potentials in Bose–Einstein condensates and other systems [38–46],
energy-time uncertainty, and superoscillations.
Acknowledgments: We would like to thank Sandu Popescu, Michael Berry, Justin Dressel, Matthew Leifer and
Jeff Tollaksen for helpful conversations as this research took shape. This research was supported (in part) by
the Fetzer-Franklin Fund of the John E. Fetzer Memorial Trust. The authors would also like to acknowledge
the use of the Samueli Laboratory in Computational Sciences in the Schmid College of Science and Technology,
Chapman University, for the computers we used for the simulation.
Author Contributions: Mordecai Waegell and Yakir Aharonov developed the theory. Mordecai Waegell
conceived of and designed the numerical simulation. Mordecai Waegell and Taylor Lee Patti ran the simulations.
Mordecai Waegell and Taylor Lee Patti analyzed the data. Mordecai Waegell and Taylor Lee Patti wrote the paper.
All authors have read and approved the final manuscript.
Conflicts of Interest: The authors declare no conflict of interest.
Appendix: Numerical Simulation
In order to characterize the effect of rapidly raising a potential barrier within the infinite square
well, we numerically solve the time-dependent Schrödinger equation during the period τ that the
potential is changing [38,47]. We used a modified version of the fourth order Runge–Kutta method,
where the modification is a first order approximation in the size of the time steps ∆t used in the
simulation. This approximation leads to an update at each time step of,
ψ(x, t + ∆t) = ψ(x, t) + i∆t (
1
∂2
− [V(x, t) + 4V(x, t + ∆t/2) + V(x, t + ∆t)]) ψ(x, t),
∂x2 6
(37)
where we are working in units with h̄ = 1 and 2M = 1. The first order approximation is
validated by the fact that for the range of parameters values for which we run the simulation,
2
∂
1
∣∆t ( ∂x
2 − 6 [V(x, t) + 4V(x, t + ∆t/2) + V(x, t + ∆t)]) ψ(x, t)∣ ≪ ∣ψ(x, t)∣.
The magnitude of ψ is so large compared to the correction that significant numerical round-off
error can be introduced. To avoid this, the correction terms of different magnitudes are accumulated
as separate variables in the simulation and added together freshly in order to compute the
correction at the each time step. This also allows precise computation of the average energy change,
as shown below.
For the time-dependent barrier, we use a linear rate of increase and a normalized dimensionless
Gaussian kernel of full width w at half-maximum,
Gw (x) =
e
−4(ln 2)(
x−x0 2
w )
L −4(ln 2)(
∫0 e
x−x0 2
w )
,
dx
(38)
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which reduces to the usual general form,
√
2
Gw (x) ≈
w
2
0
ln 2 −4(ln 2)( x−x
w ) ,
e
π
(39)
for narrow widths. This last form also becomes the Dirac delta-function in the limit that w → 0. We
define the parameter Aw as the overlap between the probability distribution of the initial state ψ(x, 0)
and the normalized barrier kernel,
L
Aw = ∫
0
∣ψ(x, 0)∣2 Gw (x)dx,
(40)
t
Vm Gw (x),
τ
(41)
which we will use later.
The potential is then,
Vw (x, t) =
2
h̄
for 0 ≤ t ≤ τ, and Vm = 104 2ML
2 . As discussed above, this Vm is more than sufficient to cause the
desired splitting of the lowest energy levels and eigenstates.
We set x0 = 83 L and L = 1, and divided the simulation into 103 time steps, which gives ∆t = 10−3 τ.
We performed the simulation using the method of lines, with a mesh of x values from zero to one with
step size ∆x = 10−5 L. The simulation preserves the normalization of ψ(x, t) to very high precision,
with the largest normalization error on the order of 10−9 . This serves as an estimate of the numerical
tolerance of the simulation and provides some verification that it is working correctly.
At the end of the model, the original state ψ0 (x) ≡ ψ(x, 0) has evolved into the state
ψ(x, τ) = ψ(x, 0) + ∆ψ(x). At time t = 0, the expectation value of the potential energy is
2
L
L
h̄
∗
′′
⟨V0 ⟩ = ∫0 ∣ψ(x, 0)∣2 V(x, 0)dx = 0, and kinetic energy is ⟨K0 ⟩ = 2m
∫0 ψ (x, 0)ψ (x, 0)dx.
At time t = τ, the expectation value of the potential energy is,
L
⟨Vτ ⟩ = ∫
0
(∣ψ0 (x)∣2 + ∣∆ψ(x)∣2 + ψ0∗ (x)∆ψ(x) + ∆ψ∗ (x)ψ0 (x)) V(x, τ)dx,
(42)
which is also equal to the change in the potential energy. We separate this into different terms:
L
∆⟨V⟩ψ0 = Vm ∫0 ∣ψ(x, 0)∣2 Gw (x)dx = Aw Vm is the change in potential energy simply due to the “lifting”
of the initial state. Note that ∆⟨V⟩ψ0 clearly approaches zero for a very narrow barrier centered at a
zero of the ψ(x, 0). The last three terms in Equation (42) are due to the change in the state, and we call
the sum of these terms ∆⟨V⟩∆ψ .
At time t = τ, the expectation value of the kinetic energy is,
⟨Kτ ⟩ = −
−
L
h̄2
∗
′′
∫ ψ0 (x)ψ0 (x)dx
2m 0
L
h̄2
∗
′′
∗
′′
∗
′′
∫ (∆ψ (x)∆ψ (x) + ψ0 (x)∆ψ (x) + ∆ψ (x)ψ0 (x)) dx.
2m 0
(43)
(44)
In this case, the first term is simply the initial kinetic energy ⟨K0 ⟩, and the other three terms are
the change in the average kinetic energy ∆⟨K⟩.
We ran the simulation for a wide range of dimensionless parameters (shown here with
corresponding physical units). The domain of the search was all combinations of the following
choices of initial parameters:
√
√
√
√
πx0
2
πx
2
2πx
2
3πx
2
2πx
(α sin πx
sin
,
sin
,
=
• ψ0 =
L
L
L
L
L sin L ,
L ± sin L ) (with α ≡ 2 cos L
L(α2 +1)
√ √
2 − 2)
• τ = {10−10 , 10−11 , 10−12 , 10−13 , 10−14 }[2ML2 /h̄]
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w
=
{(1, 2, 3, 4, 5, 6, 7, 8, 9) × 10−4 , (1, 2, 3, 4, 5, 6, 7, 8, 9) × 10−3 , (1, 2, 3, 4, 5, 6, 7, 8, 9) ×
−2
10 , (1, 2, 3, 4, 5, 6, 7, 8, 9) × 10−1 , (1, 2, 3, 4, 5, 6, 7, 8, 9, 10)}[L]
The two ψ0 ’s denoted by the ± are the state with a zero at the center of the barrier and its
reflection, which is maximum at the center of the barrier. The narrowest barrier width we can
reasonably simulate with a mesh spacing of 10−5 is w = 10−4 , while for w = 10, the barrier is
approximately flat, and the bottom of the entire well is raised uniformly.
We do not have analytic forms for ∆⟨V⟩∆ψ or ∆⟨K⟩ for all choices of the parameters Vm , w, τ
and ψ0 in the simulation, but we were able to deduce reasonably good fits for the narrow-barrier
regime (w ≤ 10−2 [L]) by guessing that these would be separable into a product of functions of each
parameter individually. In this regime, the dependence on ψ0 can be reduced to a simple dependence
on the derived quantity Aw , but for larger widths, there is more direct dependence on ψ0 .
The forms we obtain for the separate functions are somewhat odd, but they are in terms
of dimensionless quantities and provide reasonable fits to the simulated data. We found using
logarithmic analysis that the function obeys power laws in its various parameters. We do not ascribe
any analytic meaning to these forms.
The fit functions in terms of the dimensionless parameters are,
∆⟨K⟩fit (τ, w, ψ0 , V) = CK
Aw τ 2 V 4
,
w3
(45)
∆⟨V⟩fit
∆ψ (τ, w, ψ0 , V) = CV
Aw τ 2 V p
,
wq
(46)
with CK = 4.9895 × 10−9 and
with CV = −1.5921 × 10−7 , p = 4.3164 and q = 2.3146. These functions also show the time dependence
through V(t) = τt Vm . Because they range over many order of magnitude, we compute the error
of these functions by finding the root-mean-square error between the logarithms of the simulation
data and the fit for all combinations of parameters in the narrow-barrier regime. This then gives us
the relative errors, δ∆⟨K⟩fit = 16.21% and δ∆⟨V⟩fit
∆ψ = 57.05%. Clearly, the fit is much better for the
kinetic energy, but in both cases, we see that the fit functions provide good estimates of the orders of
magnitude of ∆⟨K⟩ and ∆⟨V⟩∆ψ , which is sufficient for our purpose here.
It is important to note that in order to use this fit function, one must put w into units [L], τ in
units of [2ML2 /h̄] and V in units of [h̄2 /2ML2 ], and the output will be in units of [h̄2 /2ML2 ]. The
numerical values are then dimensionless.
We interpret the positive curve in ∆⟨K⟩ as the barrier goes up as kinetic energy imparted to the
particle by the barrier. This kinetic energy appears as the wavefunction is pushed away from the
barrier on both sides, and this also results in the negative curve for ∆⟨V⟩∆ψ .
Overall, the simulation shows that any change in the potential, performed sufficiently quickly,
will have a negligible effect on the wavefunction or its energy. The change in energy is actually largest
for the narrowest barriers and goes to zero in the wide limit, where the entire flat bottom of the well is
being raised. We see that in order to minimize the change in energy, narrower barriers must be raised
faster and located at zeros of ψ(x) where Aw will be minimized. It is important to note that for the
state in Equation (3), with a zero at x0 , we find that Aw ∝ w2 , and so, even in this case, the energy
change appears to go to infinity as w → 0 for a finite τ. In principle, one can split a state with a wide
flat zero, for which Aw = 0 up to some w, but such a state would contain significant contributions from
high-energy modes and may fall outside the regime where our first order approximation in time steps
is valid.
As discussed above, our simulation shows that it is quite possible to accomplish the desired
splitting of the lowest energy levels of the well without significantly altering the kinetic energy of the
state, by using Gaussian barriers with physically-plausible widths and speeds. In general, the change
Entropy 2016, 18, 149
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in the wavefunction is very small, and by far, the largest effect on the energy of the state is due to the
“lifting” of the original state, ∆⟨V⟩ψ0 = Aw Vm .
For the example discussed above, with w = 10−3 [L], Vm = 104 [h̄2 /2ML2 ], τ = 10−10 [2ML2 /h̄] and
⟨K0 ⟩ = 2.8918π 2 [h̄2 /2ML2 ], the simulation returns,
∆⟨K⟩/⟨K0 ⟩ ≈ 1.58 × 10−10 ,
(47)
∆⟨V⟩∆ψ /⟨K0 ⟩ ≈ −5.19 × 10−10 ,
(48)
∆⟨V⟩ψ0 /⟨K0 ⟩ ≈ 2.29 × 10−3 .
(49)
and:
Thus, it is clear that ∆ψ, and thus, ∆⟨K⟩ and ∆⟨V⟩∆ψ are negligible. There is a noticeable
(but small) increase in potential energy, ∆⟨V⟩ψ0 , due to the lifting of the initial wavefunction. This
can be treated as a correction when considering conservation of energy; the kinetic energy of the
wavefunction is our primary interest.
We conclude by noting that the simulation we developed can be applied with many other choices
of parameters. In particular, we could use many different barrier shapes other than Gaussian, and the
barrier need not be raised linearly in time. We could also consider a much broader set of initial
wavefunctions. Our goal here was to show the behavior as the barrier approaches a delta function at
a zero of the wavefunction, and we believe that the parameter set we used and the fits we obtained
for the narrow-barrier regime were quite sufficient to that task.
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